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Questions and Answers
If f(x) = 2x^2 + 3x  1 and g(x) = x^2  2x, what is the value of (f  g)(2)?
What is the range of the function f(x) = 2sin(x) + 1?
If a circle has its center at (3, 4) and passes through the point (5, 6), what is its radius?
What is the sum of the first 10 terms of the sequence an = 3n  2?
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If f(x) = x^2 and g(x) = 2x + 1, what is the value of (f ∘ g)(1)?
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What is the equation of the line that passes through the points (2, 3) and (1, 4)?
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If f(x) = sin(x) and g(x) = cos(x), what is the value of (f + g)'(π/4)?
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What is the equation of the circle with center (2, 3) and radius 4?
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If a series converges by the ratio test, what can be said about the absolute value of the common ratio?
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What is the equation of the graph of f(x) = x^2, shifted 3 units to the right and 2 units down?
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Study Notes
Functions

Domain and Range:
 Domain: set of all input values (x) for which the function is defined
 Range: set of all output values (y) that the function can produce

Function Operations:
 Sum/Difference: (f ± g)(x) = f(x) ± g(x)
 Product: (fg)(x) = f(x) * g(x)
 Composition: (f ∘ g)(x) = f(g(x))

Function Properties:
 Even/Odd: f(x) = f(x) or f(x) = f(x)
 Increasing/Decreasing: f(x) ≥ f(y) or f(x) ≤ f(y) for x > y
Graphing

Graph Types:
 Linear: straight line, y = mx + b
 Quadratic: parabola, y = ax^2 + bx + c
 Exponential: y = a * b^x
 Trigonometric: y = a * sin(x) or y = a * cos(x)

Graph Transformations:
 Vertical Shift: y = f(x) ± k
 Horizontal Shift: y = f(x ± h)
 Reflection: y = f(x) or y = f(x)

Graphing Techniques:
 Plotting points
 Using symmetry
 Identifying asymptotes
Trigonometry

Angles and Triangles:
 Degree measure: 0° ≤ θ ≤ 360°
 Radian measure: 0 ≤ θ ≤ 2π
 Pythagorean Identity: sin^2(θ) + cos^2(θ) = 1

Trigonometric Functions:
 Sine: sin(θ) = opposite side / hypotenuse
 Cosine: cos(θ) = adjacent side / hypotenuse
 Tangent: tan(θ) = opposite side / adjacent side

Trigonometric Identities:
 Sum and Difference Formulas
 Double and Half Angle Formulas
Analytic Geometry

Coordinate Systems:
 Cartesian Coordinates (x, y)
 Polar Coordinates (r, θ)

Equations of Lines:
 SlopeIntercept Form: y = mx + b
 PointSlope Form: y  y1 = m(x  x1)
 Standard Form: Ax + By = C

Circles and Conic Sections:
 Circle: (x  h)^2 + (y  k)^2 = r^2
 Ellipse: (x  h)^2/a^2 + (y  k)^2/b^2 = 1
 Parabola: y = a(x  h)^2 + k
Sequences and Series

Sequences:
 Arithmetic Sequence: an = a1 + (n  1)d
 Geometric Sequence: an = a1 * r^(n  1)

Series:
 Arithmetic Series: Σan = (a1 + an)/2 * n
 Geometric Series: Σan = a1 * (1  r^n) / (1  r)

Convergence Tests:
 nth Term Test
 Ratio Test
 Root Test
Functions
 Domain of a function refers to the set of all input values (x) for which the function is defined.
 Range of a function refers to the set of all output values (y) that the function can produce.
 Function operations include sum/difference, product, and composition, with formulas (f ± g)(x) = f(x) ± g(x), (fg)(x) = f(x) * g(x), and (f ∘ g)(x) = f(g(x)) respectively.
 Functions can have even or odd properties, where f(x) = f(x) or f(x) = f(x) respectively.
 Functions can also be increasing or decreasing, where f(x) ≥ f(y) or f(x) ≤ f(y) for x > y.
Graphing
 Linear graphs are straight lines represented by the equation y = mx + b.
 Quadratic graphs are parabolas represented by the equation y = ax^2 + bx + c.
 Exponential graphs are represented by the equation y = a * b^x.
 Trigonometric graphs are represented by the equations y = a * sin(x) or y = a * cos(x).
 Graph transformations can be vertical shifts (y = f(x) ± k), horizontal shifts (y = f(x ± h)), or reflections (y = f(x) or y = f(x)).
 Graphing techniques include plotting points, using symmetry, and identifying asymptotes.
Trigonometry
 Angles can be measured in degrees (0° ≤ θ ≤ 360°) or radians (0 ≤ θ ≤ 2π).
 The Pythagorean Identity is sin^2(θ) + cos^2(θ) = 1.
 Sine (sin), cosine (cos), and tangent (tan) are trigonometric functions, where sin(θ) = opposite side / hypotenuse, cos(θ) = adjacent side / hypotenuse, and tan(θ) = opposite side / adjacent side.
 Trigonometric identities include sum and difference formulas, and double and half angle formulas.
Analytic Geometry
 Coordinate systems can be Cartesian (x, y) or polar (r, θ).
 Equations of lines can be in slopeintercept form (y = mx + b), pointslope form (y  y1 = m(x  x1)), or standard form (Ax + By = C).
 Circles can be represented by the equation (x  h)^2 + (y  k)^2 = r^2.
 Ellipses and parabolas are types of conic sections, represented by the equations (x  h)^2/a^2 + (y  k)^2/b^2 = 1 and y = a(x  h)^2 + k respectively.
Sequences and Series
 Arithmetic sequences have the form an = a1 + (n  1)d, while geometric sequences have the form an = a1 * r^(n  1).
 Arithmetic series have the formula Σan = (a1 + an)/2 * n, while geometric series have the formula Σan = a1 * (1  r^n) / (1  r).
 Convergence tests for series include the nth term test, ratio test, and root test.
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Test your knowledge of functions and graphing in algebra, including domain and range, function operations, and properties.